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2010 Stochastic Domination for the Ising and Fuzzy Potts Models
Marcus Warfheimer
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Electron. J. Probab. 15: 1802-1824 (2010). DOI: 10.1214/EJP.v15-820


We discuss various aspects concerning stochastic domination for the Ising model and the fuzzy Potts model. We begin by considering the Ising model on the homogeneous tree of degree $d$, $\mathbb{T}^d$. For given interaction parameters $J_1$, $J_2>0$ and external field $h_1\in\mathbb{R}$, we compute the smallest external field $\tilde{h}$ such that the plus measure with parameters $J_2$ and $h$ dominates the plus measure with parameters $J_1$ and $h_1$ for all $h\geq\tilde{h}$. Moreover, we discuss continuity of $\tilde{h}$ with respect to the three parameters $J_1$, $J_2$, $h_1$ and also how the plus measures are stochastically ordered in the interaction parameter for a fixed external field. Next, we consider the fuzzy Potts model and prove that on $\mathbb{Z}^d$ the fuzzy Potts measures dominate the same set of product measures while on $\mathbb{T}^d$, for certain parameter values, the free and minus fuzzy Potts measures dominate different product measures


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Marcus Warfheimer. "Stochastic Domination for the Ising and Fuzzy Potts Models." Electron. J. Probab. 15 1802 - 1824, 2010.


Accepted: 14 November 2010; Published: 2010
First available in Project Euclid: 1 June 2016

zbMATH: 1227.60110
MathSciNet: MR2738339
Digital Object Identifier: 10.1214/EJP.v15-820

Primary: 60K35

Keywords: domination of product measures , fuzzy Potts model , Ising model , stochastic domination

Vol.15 • 2010
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